This paper is a continuation of Zhang [M. Zhang, Continuity in weak topology: Higher order linear systems of ODE, Sci. China Ser. A 51 (2008) 1036-1058; M. Zhang, Extremal values of smallest eigenvalues of Hill's operators with potentials in L 1 balls, J. Differential Equations 246 (2009) 4188-4220]. Given a potential q ∈ L p ([0, 1], R), p ∈ [1, ∞]. We use λ m (q) to denote the mth Dirichlet eigenvalue of the Sturm-Liouville operator with potential q(t), where m ∈ N. The minimal value L m,p (r) and the maximal value M m,p (r) of λ m (q) with potentials q in the L p ball of radiusr are well defined. In this paper, we will exploit the continuity of λ m (q) in q with weak topologies and the variational method to give characterizations of L m,p (r) and M m,p (r) when p ∈ (1, ∞). By using the limiting approach as p ↓ 1, we find that the most important extremal values L m,1 (r) and M m,1 (r) can be evaluated explicitly using some elementary functions of r. The corresponding extremal problems for Neumann eigenvalues and some periodic eigenvalues will be reduced to L m,p (r) and M m,p (r).

We show that any sufficiently (finitely) smooth Z 2symmetric strictly convex domain sufficiently close to a circle is dynamically spectrally rigid, i.e. all deformations among domains in the same class which preserve the length of all periodic orbits of the associated billiard flow must necessarily be isometric deformations. This gives a partial answer to a question of P. Sarnak (see [22]).2 Remarkably, Sunada (see [23]) exhibits isospectral sets (i.e. sets of isospectral manifolds) of arbitrarily large cardinality.3 Results of this kind are usually referred to as infinitesimal spectral rigidity.

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